Exploring the Quantification of 'All' with Fibonacci Numbers

In summary, the conversation was about the wonders of Fibonacci numbers and the golden ratio, and whether or not certain properties can be said to be true for all numbers. The speaker believes that some properties can be proven to be true for all numbers, but their friend disagrees, stating that the word "all" quantifies an infinite set of numbers. The conversation will continue in a few days.
  • #1
Imparcticle
573
4
Yesterday I was enlightening a friend of mine concerning the many wonders of Fibonacci numbers and the golden ratio (let this friend be represented as X). As I was speaking with X, I learned that another friend of mine (let him be represented as V) was listening very attentively. Here is our conversation, as it will make things easier for explanation:

Me: As you can see, X, phi (in the form 1.6...) is the only number whose
square is (phi - 1). No other number, as far as I know, has this quality. Apparently, this is supposed to be true for all numbers...

Friend V: No. You can't say "for all numbers".

Me: Ah, because by saying "all" I am quantifying an infinite set of numbers?

Friend V: Yes.


Unfortunately, we were unable to continue this conversation on that day. We are scheduled to continue a few days from now.
I actually disagree with V's assertion. This is because of a simple fact: there are certain properties of numbers that have been proved to be true for all numbers (right? :rolleyes: ). So I could say "all numbers".
But can I say "all of the subsets of an infinite set are such that x is always true for all of the subsets"?
is my usage of the word "all" quantifying my subject? What does the word "all" do in terms of how it quantifies?
 
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  • #2
x^2 = x + 1
x^2 - x - 1 = 0
(1 +- sqr(1 - 4*1*=1))/2*1 = 1/2 +- sqr(5)/2 = 1/2 +- sqr(5/4)
= 1.618 or -.618

(-.618)^2 =~ .381
-.618 + 1 =~ .382

Hmmm
 
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  • #3
wow. thanks for the clarification.
 
  • #4
Well, what you said is badly phrased, but that's what happens with spoken English, and as alkatran shows incorrect, but it is perfectly possible to make a statement about something being true for an infinite set.

x^2>x for all x in (the infinite set) (1,infinity)

something is true for all elements in some set if the negation, that there is *an* element for which it is false, is false.

Something is true for all the quantified members to which it applies if it is, erm, true for them all.
 

Related to Exploring the Quantification of 'All' with Fibonacci Numbers

1. What are Fibonacci numbers?

Fibonacci numbers are a sequence of numbers where each number is the sum of the two preceding numbers, starting with 0 and 1. The sequence goes like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. This sequence was first described by Leonardo Fibonacci in the 12th century, but it has been found to occur in many natural and mathematical phenomena.

2. How are Fibonacci numbers used in quantification?

Fibonacci numbers can be used in quantification by assigning each number in the sequence to represent a specific value or quantity. For example, the number 1 can represent one unit, while the number 2 can represent two units. This can be applied to various measurements and calculations, making it a useful tool in quantification.

3. Can Fibonacci numbers be used to quantify 'all'?

Yes, Fibonacci numbers can be used to quantify 'all' by assigning a number from the sequence to represent a larger quantity. For instance, the number 13 can represent 13 units, while the number 21 can represent 21 units. This can be applied to a wide range of scenarios, such as counting objects, measuring distances, or even estimating populations.

4. Are there any limitations to using Fibonacci numbers in quantification?

While Fibonacci numbers can be a useful tool in quantification, there are some limitations to consider. One limitation is that the sequence is infinite, so it may not be practical to use extremely large numbers for quantification. Additionally, as with any mathematical tool, it is important to use it appropriately and understand its limitations in order to avoid incorrect or misleading results.

5. How can exploring the quantification of 'all' with Fibonacci numbers benefit scientific research?

Exploring the quantification of 'all' with Fibonacci numbers can benefit scientific research in several ways. It can provide a unique perspective on quantifying large or complex quantities, and it can also be used to estimate unknown quantities or make predictions. Additionally, incorporating Fibonacci numbers into research can lead to new insights and discoveries in various fields, such as biology, economics, and computer science.

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